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Lotka Volterra Model
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The environmental system model is based on a Lotka Volterra type model in which environmental variables are included.

The model considers a series of species that interact with each other and environmental conditions that can favor or harm their development.

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Environmental model
Equation

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If the generalized Lotka Volterra model is combined

\displaystyle\frac{d n_i }{dt}= r_i n_i + \displaystyle\sum_j \alpha_{ji} n_i n_j



with model for ambient effect

r_i = \beta_i + \displaystyle\sum_k^n ( \gamma_{ki} e_k + \delta_{ki} e_k ^2)



an environmental model governed by the equation

\displaystyle\frac{d n_i }{dt}=( \beta_i + \displaystyle\sum_k( \gamma_{ki} e_k + \delta_{ki} e_k ^2)) n_i + \displaystyle\sum_j \alpha_{ji} n_i n_j

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Generalization of the r_i factor of the Lotka Volterra model
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The factor r_i represents the way in which the species develops in the event that no other species exists.

If the factor r_i is negative the population decreases while if it is positive it grows exponentially. In the first case it is assumed that it does not have the resources so that it has to procreate less than what is required to maintain the population. In the second case, there are plenty of resources and the population expands without control.

Whether or not the necessary resources exist will depend on n environmental factors that we can define as e_k. The factor r_i will be a function of these and if it is assumed that it can be expanded in these up to the second order, the relation

r_i = \beta_i + \displaystyle\sum_k^n ( \gamma_{ki} e_k + \delta_{ki} e_k ^2)

The factor \beta_i is the sum of all the constant factors of all expansions.

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Generalization of the Lotka Volterra model
Equation

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If the Lotka Volterra model is generalized to N species, the equations

\displaystyle\frac{d n_1 }{d t }= r_1 n_1 + \alpha_{12} n_1 n_2



and

\displaystyle\frac{d n_2 }{d t }= r_2 n_2 + \alpha_{21} n_2 n_1



can be written as

\displaystyle\frac{d n_i }{dt}= r_i n_i + \displaystyle\sum_j \alpha_{ji} n_i n_j

where n_i is the population of the i th species, r_i is the growth factor of the species and \alpha_{ij}< /tex> the interaction factor.

By way of generalization, we can leave the diagonal factor \alpha_{ii} which would end up modeling competition within the same species.

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